Melvyn Klein

Abstract

Let f(z) be a member of the
family S of functions regular and univalent in the open unit disk whose Taylor
expansion is of the form: f(z) = z + a2z2+⋯ . Let Dw be the image of the unit disk
under the mapping: w = f(z). An inequality for the transfinite diameter of
n compact sets in the plane {Ti}1n is established, generalizing a result of
Renngli:

This inequality is applied to derive covering theorems for Dw relative to a class of
curves issuing from w = 0, arcs on the circle: |w| = R as well as other point
sets.